Ground states for a linearly coupled system of Schr\"odinger equations on $\mathbb{R}^{N}$

TL;DR

This paper investigates the existence of positive ground state solutions for a class of linearly coupled Schrödinger systems on brbr^N, analyzing subcritical, critical, and supercritical cases with variational methods and Pohozaev identities.

Contribution

It provides new existence results for ground states in coupled Schrödinger systems with periodic potentials, covering subcritical, critical, and supercritical nonlinearities.

Findings

Existence of positive ground states in subcritical case for all brbrbrbrbr brbrbrbr brbrbrbr.

Existence of ground states in critical case for brbrbrbr brbrbrbrbr brbrbrbr brbrbrbr brbrbrbr brbrbrbr brbrbrbr.

No positive solutions exist in the supercritical case.

Abstract

We study the following class of linearly coupled Schr\"{o}dinger elliptic systems $$\left\{ \begin{array}{lr} -\Delta u+V_{1}(x)u=\mu|u|^{p-2}u+\lambda(x)v, & \quad x\in\mathbb{R}^{N}, \\ -\Delta v+V_{2}(x)v=|v|^{q-2}v+\lambda(x)u, & x\in\mathbb{R}^{N}, \end{array} \right. $$ where $N\geq3$, $2<p\leq q\leq 2^{*}=2N/(N-2)$ and $\mu\geq0$. We consider nonnegative potentials periodic or asymptotically periodic which are related with the coupling term $\lambda(x)$ by the assumption $|\lambda(x)|\leq\delta\sqrt{V_{1}(x)V_{2}(x)}$, for some $0<\delta<1$. We deal with three cases: Firstly, we study the subcritical case, $2<p\leq q<2^{*}$, and we prove the existence of positive ground state for all parameter $\mu\geq0$. Secondly, we consider the critical case, $2<p<q=2^{*}$, and we prove that there exists $\mu_{0}>0$ such that the coupled system possesses positive ground state solution for all $\mu\geq\mu_{0}$. In these cases, we use a minimization method based on Nehari manifold. Finally, we consider the case $p=q=2^{*}$, and we prove that the coupled system has no positive solutions. For that matter, we use a Pohozaev identity type.